User blog:Syst3ms/Funky Notation
the name is indeed based off Funky Kong Hey it's me again. CAN was decent, but it was basically a HAN ripoff whose definition didn't even work properly along with some mere recursive extensions added on top of it. I believe the the limit of the notation introduced here outgrows all functions provably recursive in \(\text{KP}\omega\). This definition is reminiscent of pDAN, mainly because of the nesting system. Definition : Definition of arrays : \(m\) is an array, where \(m\) is any natural number \(\#\) is an array, where \(\#\) is an array. We call # a nested array. \(\#Am\) is an array, where \(m\) is a natural number, \(A\) is a separator and \(\#\) is an array \(\#A\#'\) is an array, where \(A\) is a separator and \(\#\) and \(\#'\) are arrays. We call #' a nested array. Definition of separators : \(\{m\}\) is a separator, where \(m\) is any natural number \(\{\#\}\) is a separator, where \(\#\) is an array. We call # a nested array. \(\{\#Am\}\) is a separator, where \(m\) is a natural number, \(A\) is a separator and \(\#\) is an array \(\{\#A\#'\}\) is a separator, where \(A\) is a separator and \(\#\) and \(\#'\) are arrays. We call #' a nested array. Note : The comma is a shorthand for \(\{0\}\) Rules : Base rule : \(a0 = a+1\) Trailing rule : \(aA\enspace 0 = a\#\) and \(\{\#\enspace A\enspace 0\} = \{\#\}\) Sandwich rule : \(aA\enspace 0\enspace B\enspace \#' = aB\enspace \#'\) and \(\{\#\enspace A\enspace 0\enspace B\enspace \#'\} = \{\#\enspace B\enspace \#'\}\) iff \(L(A) Process : Let \(\text{ad}\) and \(\text{rad}\) = 1 Let \(A_0\) be the main array \(a\) is always the base, and \(b\) is always the current entry If \(b = 0\), set \(\text{rad}=0\) and move to the next one. If \(b = 0\), replace it with \(a\). The process ends. If \(b\) is a pair of square brackets with a nested array that is not "\(0\)" inside it, let \(A_\text{ad}\) be the said nested array, increment both \(\text{ad}\) and \(\text{rad}\), jump inside it at its first entry, and go back to step 1. If \(b > 0\) : If \(\text{ad}=\text{rad}\) : Let \(f(n) = nB\) where \(B\) is the same as \(A_0\) except that \(b\) is decremented. Replace the entire expression with \(f^a(a)\) The process ends. Find the smallest strings X, Y, P, Q and \(c\) such that \(A_{\text{ad}-\text{rad}-1} = ``X\enspace 0\enspace \{P\enspace c\enspace Q\}\enspace b\enspace Y``\) If \(“\{P\enspace c\enspace Q\}”\) is the comma (P and Q are empty and \(c=0\)) : Replace \(A_{\text{ad}-\text{rad}-1}\) with \(``X\enspace S_a\enspace ,b-1\enspace Y``\), where \(S_{x+1} = “S_x\enspace ,b-1\enspace Y'\) and \(S_0 = ``0``\) The process ends. If \(c>0\) and P contains no numbers: Replace \(A_{\text{ad}-\text{rad}-1}\) with \(``X\enspace S_a``\), where \(S_{x+1} = ``0\{P\enspace c-1\enspace Q\}S_x``\) and \(S_0 = ``1\{P\enspace c\enspace Q\}\enspace b-1\enspace Y``\) The process ends. Otherwise, replace \(“\{P\enspace c\enspace Q\}b”\) with \(“\{P\enspace c\enspace Q\}1\{P\enspace c\enspace Q\}b-1”\) jump inside it at its first entry. Separator and array comparison process (totally not ripped from SAN) : *For any two natural numbers n and m, L(n) > L(m) iff n > m and L(n) = L(m) iff n = m *If A and B are separators, replace occurrences of \(\langle\ldots\rangle\) with \(\{\ldots\}\); if A and B are arrays, ignore them. #Let \(A = \langle a_1A_1a_2A_2\cdots a_{k-1}A_{k-1}a_k\rangle\) and \(B = \langle b_1B_1b_2B_2\cdots b_{l-1}B_{l-1}b_l\rangle\) #If \(k = 1\) and \(l > 1\), then \(L(A) < L(B)\); if \(k > 1\) and \(l = 1\), then \(L(A) > L(B)\); if \(k = l = 1\), follow step 3; if \(k > 1\) and \(l > 1\), follow steps 4 to 9 #If \(L(a_1) < L(b_1)\), then \(L(A) < L(B)\); if \(L(a_1) > L(b_1)\), then \(L(A) > L(B)\); if \(L(a_1) = L(b_1)\), then \(L(A) = L(B)\) #Let \(M(A)=\{i\in\{1,2,\cdots,k-1\}|\forall j\in\{1,2,\cdots,k-1\}(L(A_i)\ge L(A_j))\}\), and \(M(B)=\{i\in\{1,2,\cdots,l-1\}|\forall j\in\{1,2,\cdots,l-1\}(L(B_i)\ge L(B_j))\}\). #If \(L(A_{\text{maxM}(A)}) < L(B_{\text{maxM}(B)})\), then \(L(A) < L(B)\); if \(L(A_{\text{maxM}(A)}) > L(B_{\text{maxM}(B)})\), then \(L(A) > L(B)\); or else – #If \(|M(A)| < |M(B)|\), then \(L(A) < L(B)\); if \(|M(A)| > |M(B)|\), then \(L(A) > L(B)\); or else – #Let \(A = \langle\#_1\enspace A_{\text{maxM}(A)}\enspace \#_2\rangle\) and \(B = \langle\#_3\enspace B_{\text{maxM}(B)}\enspace \#_4\rangle\) #If \(L(\langle\#_2\rangle) < L(\langle\#_4\rangle)\), then\(L(A) < L(B)\); if \(L(\langle\#_2\rangle) > L(\langle\#_4\rangle)\), then \(L(A) > L(B)\); or else – #If \(L(\langle\#_1\rangle) < L(\langle\#_3\rangle)\), then \(L(A) > L(B)\); if \(L(\langle\#_1\rangle) > L(\langle\#_3\rangle)\), then \(L(A) > L(B)\); if \(L(\langle\#_1\rangle) = L(\langle\#_3\rangle)\), then \(L(A) = L(B)\) Analysis : Here I will show how some particuliar expressions expand. WIP Table of ordinals : Since there's been complaints about those being called "analysis", I'll call them what they are. If you are a formalist, feel free to disregard those, although they can provide some help about understanding how different functions compare to each other. WIP Category:Blog posts